Toward a Theory of Continuous Improvement and the Learning Curve
Author(s): Willard I. Zangwill and Paul B. Kantor
Source: Management Science, Vol. 44, No. 7, (Jul., 1998), pp. 910-920
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Toward
a
of
Theory
Improvement and
the
Continuous
Learning
Curve
Willard I. Zangwill * Paul B. Kantor
GraduateSchoolof Business, 1101 East 58th Street, University of Chicago,Chicago,Illinois 60637
Tantalus,Inc. and RUTCOR,Rutgers University,New Brunswick,New Jersey08903
ontinuous improvement (CI) unceasingly strives to improve the performance of production
service firms. The learning curve (LC) provides a means to observe and track that
tand
improvement. At present, however, the concepts of CI are abstract and imprecise and the rationale underpinning the LC is obscure. For managers to improve processes effectively, they
need a more scientific theory of CI and the LC. This paper begins to develop such a theory. Our
approach is based on learning cycles, that is, in each period management takes an action to
improve the process, observes the results, and thereby learns how to improve the process further
over time. This analysis suggests a differential equation that not only characterizes continuous
improvement but also reveals how learning might occur in the learning curve. This differential
equation might help management to evaluate the effectiveness of various procedures and to
improve and enhance industrial processes more quickly.
(ContinuousImprovement;
Improvement;
Learning;LearningCycle;LearningCurve;LearningTheory;ManagementEffectiveness;
Quality;TotalQualityManagement;ValueAdded;Deming;Lotka;Volterra)
C
1. Introduction
Continuous improvement (CI) is an array of powerful
techniques that has produced substantial improvements
in numerous companies and organizations. CI provides
perhaps the most central and universal component of
TQM (total quality management) which itself has
helped many companies achieve high quality and productivity. Despite the clear effectiveness of CI, however,
no scientific theory exists to guide its application or to
systematically improve the concepts of CI themselves.
This paper attempts to begin development of such a
theory. The approach presented may assist management to enhance industrial processes more quickly because it reorients the fundamental paradigm used to
achieve improvement.
Right at the start this paper connects the concept of
continuous improvement with the concept of the learning curve (LC). This makes sense because in the industrial context CI and the LC are different, yet are symbiotic. The LC forecasts how fast future costs will drop
as more of an item is produced. But it does not suggest
how to reduce costs nor how to reduce them faster. CI,
on the other hand, identifies what improvement to
make, and how to do that better and faster. In this sense,
CI removes some of the obscurity behind the LC and
can help management improve the LC's rate of learning.
The theory this paper develops for CI is then equally
applicable to the LC.
This paper also exposes and attacks a serious problem
with the LC. At present, to determine the amount of
learning (process improvement) management generally
uses the total cost method. That is, management examines the total cost in one period compared with the total
cost in the next period. But that approach, as we will see,
often produces significant statistical errors. This paper
suggests that a superior approach may be to monitor the
amount of cost reduction directly. The direct method
does not compare total costs, but only examines the few
aspects of the process that have been changed. Because
the direct method generally substantially reduces the
(rather large) error of the present method, it often significantly improves the value of both the LC and CI.
0025-1909/ 98/4407/ 0910$05.00
910
MANAGEMENTSCIENCE/VOl.
44, No. 7, July 1998
Copyright X) 1998, Institute for Operations Research
and the Management Sciences
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
To analyze and understand the improvement process
this paper employs the fundamental notion of the learning
cycle. Specifically, in each period management attempts
various strategies to make improvements. At the end of
the period management observes how effective these strategies were. That information is then employed to develop
better strategies for the next period. In one learning cycle,
for example, management might change the software parameters on a machine, see if production improves, and if
so, change the software parameters on all similar machines. Learning cycles and their inherent feedback and
adaptation capabilities seem fundamental to maintaining
rapid, long term improvement.
This paper presents five postulates that seem to underlie certain types of industrial learning and that give
rise to a differential equation that seems to describe that
learning. The differential equation introduces to the LC
and CI literature the Lotka-Volterra approach of predators and prey. Here the prey are the errors, wastes, and
other inefficiencies that impair the operations of the process. The predators are management, because they are
attempting to eradicate the inefficiencies in order improve the system.
The predictor-prey approach often produces interesting results, and it does so in our case also because the
differential equation naturally produces three types of
solutions. Two are well-known learning curves: the exponential and the power law. The third solution, however, is new in this arena and might be a new type of
learning curve. The new solution can be more fundamental than the others, as (1) it seems to depict the complete elimination of unnecessary work, and (2) the other
two solution forms can be built up from this new form.
First, the paper reviews CI, a battery of procedures
known to boost efficiency, often dramatically. Next the
LC is examined as a historical antecedent that provides
a quantitative basis for CI. Finally, the paper develops
the differential equation and explore its implications.
2. Historical Review
Continuous Improvement
The management approach called "continuous improvement" raises the efficiency of many processes and
systems and is closely integrated with total quality management, just-in-time, and kaizen (largely equivalent to
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
CI). Imai (1991) states that ". . . kaizen can increase
productivity by 30, 60, or even 100 percent or more
without major capital investment." Harmon and Peterson (1990) echo this view and ". . . expect improvement of 50 to 90 percent and more." Schonberger (1986)
cites improvement by factors of 5, 10, or even 20 in manufacturing cycle time.
Two Origins of Continuous Improvement: Toyota
and Statistical Reasoning. CI traces its origins to two
major historical trends, both dating from about 1950.
The first occurred at Toyota, where Tiichi Ohno and
Shigeo Shingo conceived the just-in-time (JIT)production system (also called kanban, Japanese production,
lean production) and catalyzed a production revolution
of a magnitude similar to that of Henry Ford a generation earlier. JITpioneered the disciplined and organized
methodology that produced the impressive efficiency
gains just cited. As one example of how this is accomplished, Toyota employees conduct systematic analyses
to improve the way that they do their jobs, and they do
this every week (Adler 1991).
The second trend underpinning CI is the quality
movement and statistical reasoning, conceived in the
1920s by Shewhart. Its contemporary renaissance is often traced to W. Edwards Deming's 1950 lectures to Japanese executives, during which he highlighted the importance of data collection and of Shewhart's Plan-DoCheck-Act (PDCA) cycle (a statement in the learning
cycle approach taken in this paper).
The Legacy of Cost Analysis: The Learning Curve
The historical predecessor of our analysis of CI is the
learning curve, which monitors and forecasts improvement by a quantitative approach. We take the viewpoint
that what drives continuous improvement is some sort
of underlying learning. The LC contributes a simple
mathematical relationship between some metric (performance measure)-such as the cost, quality, or cycle
time of producing an item-and a firm's experience in
producing that item. The resulting curve of the metric
very often follows some standard form, often a power
law. This form of mathematical description (unlike
what will be developed below) has no parameters explicitly representing the policies and procedures of
management.
911
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
Let C(q) be the cost (or other metric) of manufacturing the qth item in a series and C(1) be the cost of producing the first one. Then for p, the parameter of learning, the classical power law becomes
C(q) = C(l)q-P.
(1)
Time (t) is often substituted for the quantity produced (q).
The LC has been widely employed in nearly all industries (see Dutton and Thomas 1982 and Dutton et al.
1984). Its typical use is to forecast the cost of producing
the item in the future. Because the LC is so important,
attempts have been made to place the learning curve on
theoretical foundations.' While these analyses propose
reasons why the empirically observed learning curve
might occur, they do not suggest practical procedures
by which management might systematically accelerate
the learning. Mishina (1987) punctuates this point by
noting that the causes of learning are not clear and that
"black boxes" seem to lurk behind the curve.
The result is a "dumb" learning curve that simply
records learning, where the learning just happens,
driven by an unknown "natural economic phenomenon." Indeed, traditional LC theory provides no organized way for management to improve the slope of the
LC so that learning (improvement in cost or other metric) occurs faster.
Adding Learning Cycles to the Learning Curve
What is missing from the traditional LC theory and
would help make the learning occur faster are the concepts of CI and in particular the notion of the learning
cycle. Under the learning cycle, as noted, in each period
management attempts strategies to decrease the costs
(or other metric) and if the strategy is successful adopts
it. The example cited about improvement at Toyota il'An excellent review of the learning curve is given by Muth (1986),
who also suggests his own model based on the statistical theory of
extremes. Other work reviewed in Kantor and Zangwill (1991) includes Roberts' (1983) and Sahal's (1942) probabilistic models, Venezia's (1985) optimization model, and Levy's (1965) component-based
model. More recently, Argote et al. (1990) stressed the role of forgetting in learning, while Camm et al. (1993) noted that forgetting might
be explained by variations in production rate. Adler and Clark (1991),
for data from an electronics equipment firm, identify engineering
changes and work force training as important inputs to learning.
912
lustrates how that firm applies this learning cycle concept weekly.
To apply the learning cycle notion to the LC requires
that management determine whether the strategies it
uses in a period (cycle) really help to reduce the cost.
The "obvious" way to determine this information is to
subtract the total cost of making the item at the end of
the period from the total cost at the end of the previous
period. The amount of improvement is thus
AC(q) = C(q) - C(q - 1).
(2)
3. Problems with Identifying the
Improvement
Although calculating Equation (2) appears straightforward, it is not, because as Bohn (1991) stresses, the usual
cost data are extremely noisy.2 To illustrate, suppose the
cost in a period is 100 with an error of ?+10(standard
deviation of 10). Let the cost in the next period be 95
with an error of ?10. Using basic subtraction, the resulting cost improvement is 5. That result, however,
generally is not correct.
First consider the common case where the distributions of the errors are normal and independent. Then
the improvement is 5 ? 14. Here, although the initial
error was about 10 percent, after the subtraction to calculate the improvement, the error has ballooned to 280
percent. Next consider a most auspicious case when the
errors have a correlation of 0.5. The error is still a sizable
?10, meaning that the amount of improvement would
have a 200-percent error. Moreover, these examples are
not anomalies. An extremely large error will nearly always result because the subtraction involves two large
numbers that are similar in value. When it is measured
as a difference, the value of the real improvement is
almost anyone's guess.
The large error in the traditional approach makes it
difficult to identify statistically significant relationships
among variables-a problem that may help explain
why there has been little progress in determining an
underlying understanding of CI and LC.
2 For particular techniques to analyze the data see Box and Tiao "In-
tervention Analysis with Applications to Economic and Environmental Problems," J. AmericanStatisticalAssociation,70 (1975), 70-79.
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
Analysis of Errors
To analyze the cause of the errors, note that the total
cost (or other metric) of producing an item is actually
the sum of the costs of numerous small activities, say
a,, . .., an, each with its own random variation. These
small activities might be: entering sales data, billing,
contacting clients, training, telephoning, heat etc. Assuming n = 100 activities, the total cost in the period t
- 1 (using time instead of cumulative quantity made)
can be written
C(t - 1) = a, + a2+ a3 +
+ aloo
Suppose that during period t management attempts to
reduce the cost of some activities. Let bi denote the cost
of those particular activities at the end of the period. If
management works on two activities during the period,
the cost of activity i = 1, 2 is cut from ai to bi. Since
management does not work on any of the other activities, their costs should be unchanged except for random
disturbances. Let those costs at the end of period t be
designated a*, ... , a 0o.The total cost in period t is thus
C(t) = b, + b2 + a3*+
+ a*00
Calculating the improvement, AC, from Equation (2),
AC(t) = C(t) - C(t - 1)
= [a, + a2 +
a3 + * * * +
a100]
+
(3)
+ a*o].
-[b,+ b2 + a3*
Using the approach of subtracting total costs (Equation
(2)) the improvement calculated includes the errors of
200 cost estimates.
Some observers have suggested that the error might
be reduced by taking a long series of observations. But
that approach does not help because it takes too much
time to obtain the data, perhaps quite a few learning
cycles. Our goal is to obtain information quickly so we
can rapidly learn how to make better improvements.
To overcome some of these error difficulties, we propose to measure the amount of cost improvement by
looking only at the activities improved. In the above
example, since only two of the 100 activities were
changed, we can calculate the improvement using the
equation
Since this equation has the errors of only four terms to
contend with, the error is substantially reduced.3
Example Of Obtaining Data
But how can we obtain the data needed for Equation
(4)? To illustrate this, suppose for action 1 management
moves machine A next to machine B, eliminating the
need to forklift the material from A to B. Say management estimates the cost of the forklifting as $2. Moving
machine A next to B then should cut $2 from the cost of
making the item. Consequently, the quantity a, - b
= $2.
As a second action during the period, suppose management revises the software on their enterprise resource planning system. Suppose that reduces the scrap
and defect rates, thereby cutting $7 from the cost of
making the item. Hence a2 - b2= $7.
Whenever several individual improvements are
made, their sum is assumed be to the total improvement
made. Recasting Equation (4)
AC(t) = a, - b, + a2- b2= $8,
(5)
for a cost reduction of $8.
While this $8 value undoubtedly has some error, it is
likely to be a small fraction of the error produced by
subtracting the total costs as in Equation (3). This is
because the errors of only a very few terms accumulate
in Equation (5), but Equation (3) bears the error of 200
terms (assuming 100 small activities).
This approach of looking directly at the actions improved makes certain assumptions. For instance, the
improvements made should not interact, and also the
improvements made should not cause any decreased
effectiveness in other parts of the process. In practice,
these assumptions should be checked by inspection and
careful monitoring, and when necessary some or all of
the additional terms in Equation (3) might have to be
included. Also, the cost improvement of an activity
should be easily obtained. In general that can be done
by measuring the relevant operational factors before
and after the change, for instance, the reductions in
time, scrap, rework, labor, machine time, and so on. In
most cases obtaining the amount of improvement
3In fact, if activities 1 and 2 were improved by different methods, we
AC(t) = (a, - b1) + (a2 - b2).
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
(4)
would look at each difference separately.
913
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
directly is easy and inexpensive and requires minimal
data collection (Hands-On, How-To, Kaizen and JIT
1992). Unless the situation is highly unusual, only a few
of the terms in Equation (3) will be needed.
4. Psychological Learning Theory
Since learning and improvement-the central issues
here-are human endeavors, psychological learning
theory may be relevant to the model to be developed.
Newell and Rosenbloom (1981) noted that the power
law of learning (Equation (2)) empirically fit a number
of psychological learning situations. They also suggest,
however, the importance of the exponential, where for
a parameter r, the exponential law has the form
M(q) = M(O) exp(-rq).
(6)
(Here we use M(q) to unmistakably depict a general
metric, not necessarily cost.)
The exponential's use in human learning is strongly
justified from a theoretical viewpoint. Mazur and Hastie
(1978) comment, "For more than two decades psychologists have relied on exponential equations more than
any others . . ." and "reliance on the exponential has
not declined. . ."
Exponential Laws and Industrial Improvement
The exponential law is also well-known in industrial
learning situations. Levy's (1965) LC model uses the exponential, as does the analysis by Schneiderman (1988).
Muth (1986) has also noted departures from the power
law and cites several other forms that have appeared in
the literature.
A particularly interesting application of the exponential law was made at Analog Devices (see Schneiderman
1988 and Stata 1989) when they discovered that many
of their important business processes followed an exponential (or half-life law, as they called it). Such processes included on time delivery, outgoing defect levels,
lead time, manufacturing cycle time, process defect
level, yield, and time to market. Management then used
these half-life curves to promote the elimination of
problems and to enhance the rate of improvement.
Relative to the industrial and the psychological learning theory literature, the classical power law, Equation
(2), and the exponential, Equation (6), seem to be the
two curves in most widespread use.
914
5. The Differential Equation
Postulates
The differential equation is founded upon five postulates.
POSTULATE Ia. For any given metric, M, and process,
the process can be operatedso that it achieves its optimum
performancelevel, denotedby Z*, as measuredby that metric.
Also, (Ib): managementcan estimateZ*.
Given any process and the available equipment, labor, and other resources, we postulate that management
can estimate an optimal way (ideal, best) to operate the
process. The value of the metric when the process is
operating at an optimum is denoted as Z*. Depending
upon the metric Z might, for example, depict the fastest
time to make the product, or the lowest cost.
The Optimal Value Z* Has Different Definitions
Like its sister term, "quality," the optimal value Z* is
difficult to define precisely. Depending on the application, management can estimate Z* by extrapolation or
in a variety of ways (Zangwill 1993), such as:
1. Benchmark.Management can study the performance of similar systems or processes at the best firms
in the world. Management then sets the goal, Z *, so that
the system would then be operating at least as well as
the best such process.
2. TechnologicalEntitlement.The system may be evaluated by engineers to determine its performance if it
were operating optimally, with no wastes, defects, or
problems of any kind. Management then establishes this
performance level as the goal to strive for, Z*. This definition for Z* has been widely utilized at firms like Alcoa (Zangwill 1993).
EXAMPLE. To illustrate the entitlement approach for
determining Z*, consider a process for handling a sales
order, and let the metric be cycle time. Suppose the process takes three weeks, during which there are three major steps: the order itself must be checked for accuracy,
the customer's credit must be checked, and an order
must be placed for the material required to make the
item. During this process the order must be transferred
to each of the three different offices that do the respective steps (see figure 1).
In this case Z*, the minimum cycle time for this process, is determined by the "real" work. If, for example,
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
Figure1
Processinga SalesOrder
CHECK
ORDER
CHECK
CREDIT
ORDER
MATERIAL
(1/2HR)
(1/2HR)
(1/2HR)
THREE WEEKS
each of the steps takes about a half hour of someone's
time, Z* is 1.5 hours. During the rest of the time (three
weeks, less 1.5 hours), the sales order is being sent from
office to office or sits on someone's desk (or in the memory of some computer). These latter activities are examples of NVA, or waste, when no productive work is
being done while the sales order ages. In an ideal process they would not exist.
Reaching the Performance Goal
With some training, management can usually decide
upon the optimal point, Z*. Moreover, it is usually not
necessary to know Z* accurately, because the actual operation of the system is usually so far away from Z* that
a sizable error in Z* has little managerial import. For
instance, in the above example, it makes little difference
if Z * is 1.5 hours or 3 hours, since what must be attacked
is the cycle time of three weeks. As the system operates
closer to Z*, management can revise the estimate of Z*.
The usual challenge facing management is not how to
estimate Z*, but how to improve the system so that it
operates close to performance level Z*.
POSTULATEII. Other things being equal,for any metric
M(q), the rate of improvement is proportional to the
nonvalue-added(NVA) componentof the metric,N(q). Specifically,
M(q) = (otherfactors) x N(q)
Further,specifythe NVA as
N(q) = M(q) - Z*.
(7)
In common parlance NVA is frequently construed as
any workthat does not contribute value to the final customer. For mathematical precision, however, we define
NVA as in Equation 7, so that NVA is any "work" not
required by the ideal operating process Z*. In particular,
if M(q) is the current performance level, then M(q) Z* represents the amount of metric that remains to be
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
removed before the process will operate optimally. In
actual practice the common usage of NVA and our definition are almost identical. Typically, NVA includes rework, waiting, changes, delays, erroneous information,
defects, wastes, preparation time, transportation, idle
time, and inspection. Also, NVA would be the work in
making any items not sold. Postulate II can thus be interpreted as follows: If M(q) - Z* is large, there are
many opportunities to improve, so the rate of improvement should be large. Conversely, if M(q) - Z is small,
there are few opportunities and the rate of improvement
should be small.
III. The rate of improvementis proportional
to the effectivenessof managementin a Volterra-Lotkaform.
POSTULATE
In the 1920s Lotka and, separately, Volterra (see Murray 1989) suggested an equation describing populations
of predators and prey. As a simple instance, wolves
prey on moose. Over a wide range of conditions, the
rate of elimination of moose over time (- dS / dt) will be
proportional to:
(1) the number of moose S(t), as the opportunities
for each wolf are proportional to the number of moose;
(2) the number of wolves W(t), as the moose population will decline in proportion to how many wolves
are preying upon them.
This analysis led Lotka and Volterra to the multiplicative relationship:
dS / dt = -aW(t)S(t).
In the present context, the "prey" is the nonvalue added
component of the metric, N(q) = M(q) - Z*. That represents the errors, extra work and waste that management wants to get rid of. The "predator" is the management, or more precisely the effectivenessof management's effort to improvethe process. Let E(q) denote that
effectiveness of management in making improvements
after q items are made.
Consequently, we may rewrite the Lotka and Volterra
observations4 as
the rate of improvement in the metric is proportional to
4 See also Fine and Porteus (1989), who express improvement as a
random variable that is a fraction of the amount left to improve.
915
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
(i) the effectiveness of management, E(q), and
(ii) the amount of metric left to eliminate, N(q).
6. The Continuous Improvement
Equation
The postulates combine to produce our continuous improvement differential equation. Assuming that low
values of the metric are "better," and noting that Z* is
a constant,
-
dN / dq = rate of improvement in the metric.
(8)
Following Postulate III, we then have for c a coefficient
of proportionality:
dN/dq = -cE(q)N(q).
(9)
To measure improvement in any practical application, express Equation (9) in finite differences:
/N/ Aq = -cE(q)N(q).
(10)
These forms, Equation (9) and also Equation (10), are
the "continuous improvement differential (or finite difference) equation" (CIDE).
7. Monitoring the Effectiveness
of CI
In applying the CIDE to actual operations, the quantity
Z*, being the ideal performance of the process, would
be estimated early in the project. M(q) would also be
known, since that is the value of the metric, say cost,
after q items are processed. Further, c would be set to
the scale of the data, so without loss of generality we
can set c = 1. To apply the CIDE, thus, we need only
determine the rate at which the metric improves, /M /
Aq, as that will provide the effectiveness of management: E(q).
Specifically,
E(q)
AMM/
(/qN(q)).
Practical Importance of the CIDE
Using the metric M(q) alone does not easily provide
information about the effectiveness of an improvement
technique. But the CIDE does, because it normalizes the
improvement to account for the amount of metric left,
N(q) = M(q) - Z.
For example, suppose SPC is implemented on a specific machine and $3 per unit is saved, so AM1 = $3.
Suppose that later, after more products are processed, a
new incentive system motivates the workers to save $2
per item, so L\M2 = $2. Since $3 is larger than $2, it
appears that installing SPC has a bigger impact than the
new incentive system, and that SPC ought to be deployed through the entire operation before the incentive
system is. But that might not be correct.
The absolute amount of savings must be adjusted for
the amount of NVA left, N(q). In particular, suppose,
Ni(q) = $6 and N2(q) - $3. Then from the CIDE, since
$2 / $3 is greater than $3 / $6, the incentive system is actually more effective.
What is relevant here is the fraction of NVA cut, not
the absolute amount cut. Without use of the CIDE, management might erroneously replicate a method that had
been effective only because it was applied to a situation
with a sizable value of N(t). The CIDE automatically
adjusts for the NVA left and thereby produces an appropriately scaled measure of impact, the percentage of
NVA cut.
(11)
During any period of interest, Aq is the number of items
processed. (If we are examining how the metric improves from the time the 36th item is made until the
39th item is made, then /\q = 3.)
Next we must estimate AM, or how much the metric
changed during that interval. As discussed, direct ob-
916
servation is generally preferable to inference from some
change in a grand total. With that done, E(q) is then
known.
In general, suppose that during different intervals of
time a firm implements different CI techniques, such as
SPC, moving the machine, improving maintenance,
training, etc. Then for any specific CI technique implemented, E(q) expresses the corresponding effectiveness
of CI. That information lets us identify which improvement effortsor techniquesare morepowerful.
8. Autonomous Solutions of the
CIDE
So far the CIDE has been given in general form. We now
show that for simple dependencies of the E(q), effectiveness of management, on the current level of N(q),
the CIDE can be solved explicitly.
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
To motivate that expression, note that E(q) might depend upon the amount of known NVA in the process:
N(q) = M(q) - Z*. On the one hand, when there is a
compelling need to improve, such as a tight deadline,
management might increase its effort toward the end.
In this case, as M(q) - Z* gets small, E(q) increases.
On the other hand, without special pressure, as the
optimum point Z* is approached and little possible improvement remains, management might shift its attention away from this project to other more immediate
needs. In this case, as M(q) - Z* gets small, E(q) decreases. Postulate IV subsumes both of these possibilities.
Theeffectivenessof the CI effortdepends
upon the amount of improvementremaining,accordingto a
powerlaw. Specifically,for a parameter,K, and coefficientK,
POSTULATE IV.
(12)
E(q) = KN(q)K.
Equation (12) represents ways in which management
can change the level of effort (effectiveness) it puts into
this activity as improvement is accomplished. In particular, N(q) = M(q) - Z* is decreasing as improvements
are being made. If the parameter K > 0, the effectiveness
of management is decreasing, while K < 0 represents
increasing effectiveness, and K = 0 is constant effectiveness of management.
9. Explicit Solutions of the CIDE
Plugging Equation (12) into the CIDE Equation (10) we
obtain
dM/dq = -cK(M(q) -
(13)
Z*)K+l.
Solutions
Depending on the parameter, three solutions to Equation (13) result (Kantor and Zangwill 1996). If K > 0,
the solution is a power law, which, as in Equation (1),
is the classical form of production learning. Restoring
the full value of the metric, M(q), the general form is:
M(q)
=
(M(1)
-
Z*)(1
+ qoff)P(q
+ qoff)-P
+ Z*.
When the offset parameter qoff, which shifts the origin,
is set to zero, the classical law results:
M(q) = (M(1) - Z*)q-P + Z*, q ? 1.
This law is often written with Z * = 0.
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
(14)
If K
=
0, the solution is an exponentiallaw.
M(q) = (M(O) - Z*) exp(-rq) + Z*.
(15)
Setting K < 0 generates a new class of solutions,
which go to Z* at a finite amount of production qmax.
We call this class the finiteform, since it seems to represent specific categories of NVA being (totally) eliminated from the process in a finite period of time.
M(q) = (M(O) - Z*)(1
= Z* forq
- q/qmax)d
+ Z*
for q <
2 qmax.
qmax
(16)
Empirical Evidence
Certainly, the power law and the exponential are wellknown to reflect learning. More important, empirical
testing confirmed that the new form, the finite, also
seems to reflect industrial learning. Specifically, the
three forms were tested on published data series from
the quality progress literature. The data series subsumed metrics as diverse as "cost per keystroke" (for
data entry) and "errors per invoice," and included various cycle time measures. For the 21 data series analyzed, the finite form fit the data best in 9 cases, and the
power and exponential forms were each best in 6 cases.
In sum, all three forms were empirically validated and
were roughly equally good in fitting the learning and
improvement data (See Kantor and Zangwill 1996). The
empirical evidence for the three forms can be interpreted as preliminary support for the CIDE Equation
(8) and for Equation (13).
10. Interpretationof the Three
Families of Solutions
Since all three solution forms of the CIDE seem useful,
let us examine them in more detail.
Power Law and Exponential Forms of Learning
In the LC literature the classical form is the power law,
Equation (14). However, as mentioned, other equation
forms have also appeared in the literature, most notably
the exponential form, Equation (15).
Managerial Application of the Exponential
From a managerial viewpoint, the exponential is a very
attractive choice because it is memoryless. The other
forms are more cumbersome because they require
917
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
specification of a starting point, a shift q0ff,or an ending
point qmax. To apply the exponential management needs
merely to articulate the goal of achieving a given percentage improvement per unit time. Hewlett-Packard
states goals this way. At Analog Devices, Schneiderman
(1988) introduced the exponential, and it is discussed
as part of that firm' strategy by Stata (1989).
Motorola also employs the exponential in forming
goals (Smith and Zangwill 1988). The proprietary data
we obtained from Motorola, while not conclusive, indicate that exponential improvement was holding for
several factors-of-2 improvement. As a means for management to establish improvement goals, the exponential seems the form of choice.
total defect events (quality), it is the sum of all defect
events occurring in all activities of the production process, and Mi is the number of defects contributed by the
ith activity. If M(q) is the cost to make an item, then the
terms Mi (q) represent the cost of the numerous activities
in the process. For multiplicative metrics, such as the
yield in wafer production, the logarithmic transform
leads to additivity.
Exponentials Sum to Power Law
Utilizing Postulate V, the exponential and the power
law become intimately related. The power law of improvement can be represented by a sum of exponentials,
or in the limit, as the integral of exponentials.
(1?
Finite Forms
The finite form (Equation (16)) has not previously appeared in either the learning curve or CI literature. With
the exponential or power models NVA vanishes only
asymptotically. The finite form, however, goes to zero
in finite experience or time, and this seems to happen
in practice. In manufacturing, for example, a just-intime system might eliminate certain inventory storage.
Metric components associated with the storage, such as
cost, time of storage, or defects occurring during storage, then totally disappear. Thus the finite form seems
to describe the elimination of some step in the work
process.
q.in
+
)P(q
qmin )-P
0
= (1
f
+ qnin)p
1
17()
e-rqminrPe-rqdr.
Equation (18) suggests that, at least in principle, the total metric M(q) could improve as a power law, while
the component parts of that metric, Mi(q), improve as
exponentials (see Kantor and Zangwill 1991).
Finite Forms Approximate Exponential Forms
It is also true is that the exponential can be approximated as a sum of finite forms (Kantor and Zangwill
1996). The exponential can be written as:
e-rq
(1
=
-
q/qmax)d
q
11. Hierarchyof Solution Curves
d
Xqmaxr d+1
The three forms are not just solutions to the CIDE. They
also seem to form a hierarchy describing the removal of
NVA when viewed at different scales of detail. To explore this we introduce Postulate V.
POSTULATE V. Additive Decompositionof Metric. The
metricM(q) of a processcan be decomposedinto submetrics
Mi(q), which are metricsfor steps of the process,and whose
sum is the metricM(q).
M(q)
=
, Mi(q).
(17)
This interpretation holds, with appropriate amendment, for the key metrics of cycle time, quality, and cost.
For example, if M(q) represents total cycle time to complete a process, Mi(q) is the time to complete the ith step
on the critical path for that process. If M(q) represents
918
(18)
+ 1)
e ~mxq~.(19)
-rqmaxdqmax
The Hierarchy of Improvement Curves
The three functional forms appear to compose a hierarchy. Sums of the finite form can well approximate the
exponential form, and sums of exponentials can well
approximate a power law. However, the reverse is not
true, as such sums would require negative coefficients,
which seems to make no sense. Since the finite form can
generate the other two, but not conversely, the finite
form seems the most fundamental.
As mentioned, each finite term can be interpreted as
the elimination of a category of NVA. Consequently, it
might be that both the power law and the exponential
are observed at high levels of aggregation when the underlying reality is that many small contributions to the
metric are eliminated entirely.
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
ZANGWILL AND KANTOR
Towarda Theoryof ContinuousImprovementand the LearningCurve
Perspective on the Three Forms
As Feller (1940) noted, the power law is something of a
catchall, because it provides a good fit for a variety of
curves. When a power law is observed, we cannot ignore the possibility that, in reality, the data are the sum
of other forms. This could largely explain the dominant
role of the power law in the learning curve literature,
where cost data are sums of hundreds or thousands of
separate series of data. Specifically, even if the total metric, M(q), appears to follow a power law, the detailed
source metrics, Mi(q), might not. They might be finite
forms, depicting quantities of metric being eliminated.
13. Summary
Lack of a quantitative theory for the learning curve and
for the concepts of continuous improvement has inhibited their development and their application. We suggest that a powerful method to produce improvement /
learning is repeated use of the learning cycle. During
each learning cycle, management can observe what
techniques are producing greater improvement and
thus can learn how to improve processes faster and
faster.
The traditional way to obtain data for the learning
cycle would be to subtract the total cost (or other metric)
in one period from the total cost in the previous period.
That technique, however, can introduce sizable errors,
and we propose that it is often possible to reduce that
error by measuring the individual improvements directly.
The notion of the learning cycle allows us to propose
the beginnings of a theory of continuous improvement /
learning. This theory is based on a five postulates and
leads to a differential equation describing the learning.
This equation then leads us to three forms of the learning curve: the power law, the exponential, and the finite.
With regard to the coexistence of power law and exponential forms of learning, the differential equation
shows that these may belong to the same family as a
new law of learning, the finite form. In industrial situations, the power and exponential laws might be sums
of finite forms, suggesting that the finite law may be
more basic.
It is hoped that this theory will be tested through direct measurement of the elimination of NVA. Other
MANAGEMENTSCIENCE/Vol. 44, No. 7, July 1998
tests might examine whether complex systems are governed by the finite form at the level of small component
processes, by exponential forms at intermediate levels
of aggregation, and by power laws at the highest aggregation levels.5
This work has been supported in part by the National Science Foundation program in Decision, Risk, and Management Sciences grant
SES8821096 to Tantalus, Inc. The authors wish to thank Paul Noakes,
Vice President of Motorola, Inc., for his help in obtaining data for this
project. Professors Harry Roberts and Gary Eppen, Graduate School
of Business, University of Chicago, contributed valuable advice.
Moula Cherikh and Xiaomei Xu, of Tantalus, Inc., performed computer analyses. We acknowledge the thoughtful and persistent efforts
of the editors and referees.
5
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